A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
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16 views

Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
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5 views

Splitting Probability Within a Finite Time

Let $\mathcal{X}=\{x_i\}_{i=1}^{N}$ be a subset of $\mathbb{Z}$. For $j\in(1,N)$, what is the probability that the first element of $\mathcal{X}$ encountered by a simple 1D random walk is $x_j$ and ...
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11 views

“Return probability” to origin of a variant of the random walk.

Let $\{\epsilon_t\}_{t\ge0}$ be an iid sequence of random variables and let $\lambda>1$. I am interested in the following process: Let $X_0 = 0$ and $$ X_{t+1} = \lambda(X_t+\epsilon_t). $$ This ...
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7 views

Local time accumulated on an interval

On Wikipedia, the definition of local time is $$L^x(t) = \int_0^t \delta(x - B_s) ds$$ where $B_s$ is a real-valued diffusion process, and $\delta$ is the Dirac delta function. My question is, are ...
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8 views

M/G/1 queuing system with two arrivals

I have a queuing system with two independent Poisson arrivals with rates $\lambda_1$ and $\lambda_2$. But, the service time for each arrival is different. Suppose f_1(s) and f_2(s) are the pdf of ...
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26 views

ergodic theorem for expectation of positive recurrent diffusion

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{...
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25 views

Nonuniqueness of Stochastic Differential Equation

Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and $$dX_t = \mu(t,X(t)...
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17 views

Data transmission process PDF

Given the quasi-defined data transmission random process: $X(t) =\sum_{n=-\infty}^{+\infty} a_n \pi_T(t - nT)$ where $a_n$ are statistically independent RVs that can either assume the value 0 or 1 ...
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35 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
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44 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
2
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1answer
28 views

Showing a “signed Markov transition density” will lead to a trivial measure on path space.

Let for all $t>0$, $x\mapsto p(t,x)$ be a Schwartz function on $\mathbb R$, satisfying $\int_{\mathbb R}p(t,x)\mathrm dx=1$ and $\int_\mathbb{R}|p(t,x)|\mathrm dx\equiv C>1$ for all $t>0$ (so ...
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14 views

Exponential Gaussian volterra process. Close form conditional expectation?

Asssuming a probability space $(\Omega,(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ such that $(\mathcal{F}_t)_{t\geq 0}$ is generated by a Brownian motion $W_t$. We assume that $s>0$ is fixed and $t\in[...
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19 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
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22 views

Relationship Between $\mathbb{E}$(time) and $\mathbb{E}$(Repetition)

Consider aa Stochastic Process with Expected value of time of occurring =T (less than infinity). Can we deduce that Expected value of number of occurrences until time T is equal to 1?? If not, in ...
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11 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
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23 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
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1answer
54 views

Find the variance of W when given $W = x + 2y + 3z$. [closed]

x,y,z are random numbers given w = x + 2y + 3z. also given that the mean of x,y,z= 1,8,0 respectively. what is the mean of the random number w ? Assuming the Standard deviation of the random numbers x,...
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101 views
+50

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
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22 views

Doob Meyer decomposition in an exercize

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
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1answer
40 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
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24 views

Explain the orderliness of Poisson process

For an Orderly Poisson Process, events occur at distinct points and not simultaneously. However, the reverse is not necessarily true, i.e, even if the events occur at distinct points, the process may ...
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41 views

Understanding an equation

I am trying to understand an equation from the paper "Dynamic Model for generating Synthetic ECG signal" (http://web.mit.edu/~gari/www/papers/ieeetbe50p289.pdf). The equation is: $$S(f) = \frac{\...
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32 views

Differences between additive processes and Lévy processes

A real valued stochastic process $\left\{ X_{t}:\ t\in\mathbb{R}^{+}\right\} $ is termed additive if $\forall n\in\mathbb{N}$, $0\leq t_{0}<t_{1}<...<t_{n}<+\infty$, $X_{t_{0}},X_{t_{1}}-...
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81 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
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19 views

Optimal average utility of the processing network needed

In "Utility Optimal Scheduling in Processing Networks" by Michael J. Neely et al an example of processing network is provided. There are three queues ($q_1,q_2,q_3$) in the network and two processors (...
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17 views

Role of alpha-stability for subordinators

A Lévy process $\left\{ X_{t}\right\} $with values in $\mathbb{R}^{+}$ is termed a subordinator if it is a.s. increasing as a function of $t$, i.e. the map $t\mapsto X_{t}(\omega)$ is increasing for ...
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48 views

What is the uncertainty of a discrete sum given the uncertainty of an individual element?

I have a measurement $$X=\sum_{i=1}^nX_i,$$ and I am interested to know standard deviation $\sigma_X^2$ of measurement $X$, assuming I know $\sigma_i^2$, the standard deviation of all measurements $...
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1answer
25 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
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problem in concept of linear and nonlinear process [closed]

Is the nonlinear process is nonstationary process? in the other word: what is the relationship between stationary and linearity?
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37 views

Approximating Geometric Brownian Motion numerically

I am trying to generate a numerical solution to the SDE for Geometric Brownian Motion. The stochastic process is given by $S_t = \exp(\sigma W_t + \mu t)$, and by Ito's lemma, we have that the SDE is ...
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1answer
48 views

Stationary solution of a Fokker-Planck equation

I have a question that's driving me crazy. I have a Fokker-Planck equation $$\frac{\partial P}{\partial t}=x\frac{\partial P}{\partial x}+D\frac{\partial^2 P}{\partial x^2}$$ I look for the ...
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1answer
21 views

Deriving the Power Spectral Density of a Maximum Entropy Process

In Elements of Information Theory, Chapter 12, Section 6 Burg's Theorem is derived: Presented with the first $p$ values of the autocovariance function $R(k) = E[X_i X_{i+k}]$ a stochastic process ...
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35 views

Fractional powers of Markov generators

Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the ...
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62 views

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

Let$^1$ $U$ and $H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U,H)$ be nonnegative and symmetric operator on $U$ with finite trace $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $...
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1answer
71 views

The jumping times of a càdlàg process are stopping times.

Protter first proves this theorem: Let $X$ be an adapted càdlàg stochastic process, and let $A$ be a closed set. Then the random variable: $T(\omega)=\inf\{t: X_t(\omega)\in A \text{ or } ...
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17 views

First time of passage, discrete random walk with disjoint absorbing regions

I have a sum $T^i$ of zero/one Bern$(p)$ random variables $T_i$ and multiple disjoint absorbing regions, i.e. the absorbing region is a union of disjoint, closed sets: $$T^i \in \bigcup_{u \in \...
2
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1answer
24 views

A question about Poisson process?( Merged process)

Question: Alice shows up at time zero and spends her time exclusively in typing emails. The times that her emails are sent are a Poisson process with rate $λ_A$ per hour. And Bob just finished ...
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Show that a cadlag adapted martingale is a local martingale (help using DCT to show uniform integrability)

EDIT 2: With the correct definition, I think I have a proof. Want to show $\lim_{M\to\infty} \sup_t E[|X_{t\wedge n}|; |X_{t\wedge n}|\ge M]=0$. Fix $n$. Note that $\sup_t E[|X_{t\wedge n}; |X_{t\...
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58 views

Law of large numbers limit dependend on a second variable. What happens when both limits are taken at once?

The question I have is as follows. I have a i.i.d. sequence of random variables $(X^\alpha_n)_{n \in \mathbb{N}}$ with a expectation $\mathbb{E}X^\alpha$ which depend on a Markov process with a scaled ...
2
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73 views

Radon-Nikodym with respect to Stochastic Measure?

Question This question is now concerning stochastic processes. Let $(X_t)_{t\geq0}$ be defined on the probability space $(\Omega,\mathcal{F},P)$ with $\mathcal{F}_t=\sigma(X_s:s\leq t)$. Assume that ...
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90 views

Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
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14 views

Coupling a “partially” stationary process?

Take the stationary process $X$ on $\{0,1\}$ with distribution $\pi=(\pi_0,\pi_1).$ Then introduce the rates: $$ \begin{aligned} 0\rightarrow2 & \quad \text{ at rate } \quad \gamma_{02} \\ 1\...
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Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
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10 views

Composition of limits of functions | Switching limits of function

I have a question which I am having some trouble with. I have a double indexed sequence of stochastic processes (martingales in fact), denoted $X_{m,n}(t)$. Now I can prove that $\underset{m \...
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35 views

If a process is previsible, is the stopped process previsible? [closed]

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
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17 views

Given a set of 1-dimensional Brownian motions $\{a_i\}_{1\le i\le K}$, what is the average hitting time between $a_1$ and $\{a_i\}_{2\le i\le K}$?

Given $K$ Brownian motions $\{a_i(t), t\ge 0\}_{1\le i\le K}$ contained within interval $[0,s]$, the boundaries at 0 and s are reflected. Assume at time $t=0$, the initial locations are $a_1(0)\le a_2(...
2
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22 views

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
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6 views

increments compound poisson

What is the distribution of the increments of a compound poisson process and weiner process? I have $ X_{t} = X_{0} + \mu t + \sigma W_{t} + \sum_{i=0}^N N( \theta, \sigma^2 ) $ $ Y=\sum_{i=0}^N N(...
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35 views

Calculation probability of dynamic process model of capacity

I found this place really helpfull and now I got my first own question I cant solve. I want to unterstand the calculation of an Article im reading. Therefore we define a capacity process $C$ in a ...